Terminal singularities of the moduli space of curves on low degree hypersurfaces and the circle method

TL;DR

This paper investigates the singularities of moduli spaces of degree e maps from genus g curves to low degree hypersurfaces, showing they are at worst terminal for large e, using jet schemes and a circle method approach.

Contribution

It introduces a novel application of the circle method to analyze jet schemes of moduli spaces, establishing terminal singularities for large degree maps.

Findings

Moduli spaces have at worst terminal singularities for large e.

Development of a circle method tailored for jet scheme analysis.

New insights into the structure of moduli spaces of maps.

Abstract

We study the singularities of the moduli space of degree $e$ maps from smooth genus $g$ curves to an arbitrary smooth hypersurface of low degree. For $e$ large compared to $g$, we show that these moduli spaces have at worst terminal singularities. Our main approach is to study the jet schemes of these moduli spaces by developing a suitable form of the circle method.